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AP Calculus AB: 9.4.3 The Fundamental Theorem of Calculus, Part I

Mathematics12 CardsCreated 3 months ago

This section introduces Part I of the Fundamental Theorem of Calculus, which bridges the concepts of integration and differentiation. It shows that if a function is defined as an integral with a variable upper limit, its derivative is simply the integrand evaluated at that upper limit—provided the function meets continuity conditions. Through examples, it demonstrates how this theorem simplifies differentiation of integral-defined functions.

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The Fundamental Theorem of Calculus, Part I

  • Understand The Fundamental Theorem of Calculus, Part I, which links areas under curves with derivatives.

  • Apply The Fundamental Theorem of Calculus, Part I to differentiate a complicated function defined by an integral.

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Key Terms

Term
Definition

The Fundamental Theorem of Calculus, Part I

  • Understand The Fundamental Theorem of Calculus, Part I, which links areas under curves with derivatives.

  • Apply The Fundament...

note

  • The Fundamental Theorem of Calculus, Part I states that if a function f(x) is continuous on a closed interval [a, b] and then F(x) is conti...

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x) = ∫^x t+1/t-1 dt, [1,5]

F’(x) = is not defined over all of [1,5]

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫ x100 cost/√t−99 dt, [100,200]

F′(x)=cosx/√x−99 on [100,200]

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x17 cost/√t−17 dt, [17,40]

F′(x) is not defined over all of [17,40]

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x−2f(t)dt,[−2,2],where f is continuous on [−2,2].

F′(x)=f(x) on [−2,2]

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AP Calculus AB: 9.4.3 The Fundamental Theorem of Calculus, Part I
TermDefinition

The Fundamental Theorem of Calculus, Part I

  • Understand The Fundamental Theorem of Calculus, Part I, which links areas under curves with derivatives.

  • Apply The Fundamental Theorem of Calculus, Part I to differentiate a complicated function defined by an integral.

note

  • The Fundamental Theorem of Calculus, Part I states that if a function f(x) is continuous on a closed interval [a, b] and then F(x) is continuous and differentiable on [a, b], and F´(x) = f(x).

  • The Fundamental Theorem of Calculus is the fundamental link between areas under curves and derivatives of functions.

  • This example demonstrates the power of The Fundamental Theorem of Calculus, Part I. To differentiate the given complicated function F(x) directly requires first performing the integration, which itself requires a u substitution. Once integrated, taking the derivative requires using the Chain Rule.

  • All of this can be bypassed by using The Fundamental Theorem of Calculus, Part I. The derivative of the given
    function is simply the integrand evaluated at x.

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x) = ∫^x t+1/t-1 dt, [1,5]

F’(x) = is not defined over all of [1,5]

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫ x100 cost/√t−99 dt, [100,200]

F′(x)=cosx/√x−99 on [100,200]

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x17 cost/√t−17 dt, [17,40]

F′(x) is not defined over all of [17,40]

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x−2f(t)dt,[−2,2],where f is continuous on [−2,2].

F′(x)=f(x) on [−2,2]

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x2 5t^−1e^t dt, [2,8]

F′(x)=5x^−1e^x on [2,8]

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x−5 4t^−1 dt,[−5,−1]

None of the above

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x1 lnt/t dt,[1,4]

F′(x)=lnx/x on [1,4]

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x−3 (−4t^2+3t^−1 −7) dt,[−3,7]

F′(x) is not defined over all of [−3,7]

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x0 sint dt,[0,2π]

F′(x)=sinx on [0,2π]

Use the Fundamental Theorem of Calculus to find an expression for the derivative of the given function defined on the given interval, if it exists.
F(x)=∫x1 3dt, [1,1000]

F′(x)=3 on [1,1000]